The energy sword-wielding Elite introduced as a major playable character in Halo 2 is the Arbiter.
In Halo 2, the Arbiter is presented as a disgraced Sangheili commander who is given a ceremonial military role by the Covenant leadership. He is sent on dangerous missions, including the recovery of the Sacred Icon on a Halo installation. During the story, he learns that the Covenant’s religious interpretation of the rings is false and turns against the alliance’s leadership.
The character is later identified as Thel ‘Vadam, a former Supreme Commander of the Fleet of Particular Justice. His alliance with humanity and the Master Chief becomes one of the franchise’s defining developments. The Arbiter’s playable chapters also show the conflict from the perspective of the Sangheili.
“The Arbiter” is the title or office, not the character’s personal name. Thel ‘Vadam is therefore a related answer, but the question asks for the title used by the playable Elite.